# Difference between revisions of "2021 JMPSC Sprint Problems/Problem 17"

Mathdreams (talk | contribs) (→Solution) |
Tigerzhang (talk | contribs) (→Solution) |
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==Solution== | ==Solution== | ||

− | Notice that <math> | + | Notice that <math>1003 \cdot n = 1000n + 3n</math>. Since <math>1000n</math> always has <math>3</math> zeros after it, we have to make sure <math>3n</math> has <math>3</math> nonzero digits, so that the last 3 digits of the number <math>1003n</math> doesn't contain a <math>0</math>. We also need to make sure that <math>n</math> has no zeros in its own decimal representation so that <math>1000n</math> doesn't have any zeros other than the last <math>3</math> digits. The smallest number <math>n</math> that satisfies the above is <math>\frac{111}{3}=37</math>, so the answer is <math>1003 \cdot 37 = \boxed{37111}</math>. |

~Mathdreams | ~Mathdreams | ||

+ | ~edited by tigerzhang |

## Revision as of 12:53, 11 July 2021

## Problem

What is the smallest positive multiple of that has no zeros in its decimal representation?

## Solution

Notice that . Since always has zeros after it, we have to make sure has nonzero digits, so that the last 3 digits of the number doesn't contain a . We also need to make sure that has no zeros in its own decimal representation so that doesn't have any zeros other than the last digits. The smallest number that satisfies the above is , so the answer is .

~Mathdreams ~edited by tigerzhang